https://r2.erweima.ai/mathbot/image/17244083114487sw3fwv9.jpg

The problem asks for a description or visualization of the vector functions given in \(\mathbb{R}^3\). Let's go through each vector function and analyze its geometric properties in \(\mathbb{R}^3\).

### a) \( \mathbf{F}(t) = \sin(t)\mathbf{i} - \cos(t)\mathbf{j} \)
- **Description**: This is a parametric equation of a circle in the \(xy\)-plane.
- **Plane**: The vector lies entirely in the \(xy\)-plane, i.e., \(z = 0\).
- **Curve**: As \(t\) varies, the point \((\sin(t), -\cos(t))\) traces out a circle of radius 1 centered at the origin in the \(xy\)-plane.

### b) \( \mathbf{F}(t) = \left(2\cos(t)\right)\mathbf{i} + \sin(t)\mathbf{j} \)
- **Description**: This is also a parametric equation of an ellipse in the \(xy\)-plane.
- **Plane**: The vector lies in the \(xy\)-plane.
- **Curve**: As \(t\) varies, the point \((2\cos(t), \sin(t))\) traces out an ellipse with semi-major axis 2 along the \(x\)-axis and semi-minor axis 1 along the \(y\)-axis.

### c) \( \mathbf{G}(t) = e^t\mathbf{i} - t\mathbf{k} \)
- **Description**: This is a parametric equation representing a curve in 3D space.
- **Curve**: The \(x\)-coordinate increases exponentially with \(e^t\), and the \(z\)-coordinate decreases linearly with \(t\), while the \(y\)-coordinate is zero. This results in a curve that spirals upward as \(t\) increases.

### d) \( \mathbf{G}(t) = t\mathbf{i} + t^2\mathbf{j} + t^3\mathbf{k} \)
- **Description**: This is a parametric equation representing a 3D curve.
- **Curve**: The coordinates \((t, t^2, t^3)\) describe a space curve where \(x\) increases linearly, \(y\) increases quadratically, and \(z\) increases cubically with \(t\). The curve becomes increasingly steeper as \(t\) increases.

### e) \( \mathbf{G}(t) = \left(2\sin(t)\right)\mathbf{i} + \left(2\cos(t)\right)\mathbf{j} + 3\mathbf{k} \)
- **Description**: This represents a circle in the \(xy\)-plane with a constant height \(z = 3\).
- **Plane**: The vector lies on a plane parallel to the \(xy\)-plane at \(z = 3\).
- **Curve**: The point \((2\sin(t), 2\cos(t))\) traces out a circle of radius 2 centered at the origin in the \(xy\)-plane.

### f) \( \mathbf{R}(t) = \left(2\cos(t)\right)\mathbf{i} + \sin(t)\mathbf{j} - 2t\mathbf{k} \)
- **Description**: This is a parametric equation of an elliptical helix.
- **Curve**: The \(xy\)-components describe an ellipse, while the \(z\)-component decreases linearly with \(t\). This gives a helical path around the \(z\)-axis, descending as \(t\) increases.

### Summary:
- Functions (a) and (b) describe planar curves in the \(xy\)-plane.
- Functions (c) and (d) describe curves in 3D space with non-trivial changes in all coordinates.
- Functions (e) and (f) describe a circular path at a constant height and a descending elliptical helix, respectively.

Would you like more details on any of these functions or the steps involved in sketching them?

### Questions:
1. How can we generalize the concept of a helix in 3D space?
2. What are the properties of the ellipse described by \( \mathbf{F}(t) \) in part b)?
3. How does the exponential function in part c) influence the shape of the curve?
4. What is the significance of the cubic term in \( \mathbf{G}(t) \) in part d)?
5. How would the curve in part f) change if the \(z\)-term were \( +2t \) instead of \( -2t \)?

### Tip:
When analyzing vector functions, it’s helpful to break them down into their individual components and consider how each component varies with \(t\). This approach makes it easier to understand the overall geometry of the curve.

Explore parametric equations and 3D curves in \(\mathbb{R}^3\). This detailed analysis covers various vector functions describing curves such as circles, ellipses, spirals, and helices in three-dimensional space. Suitable for advanced undergraduate students and enthusiasts of mathematical visualization in 3D.

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